Replying to the first post:If you have broken the ribbon cable, which is highly likely if you have broken the hinge, then the DS will not work. It does not turn on unless there is a connection to both screens. You will need to buy a new ribbon cable for it:A useful site I found is: www.consoleparts4u.co.uk

Once you replace this then it should work for you.

Repling to th second post:

I would recommend buying a new case for it at least because the ribbon cable will break if left broken due to the strain on it.

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It could also be a bad top screen due to a bad ribbon cable. If your hinge is cracked it could cause damage to the ribbon cable that passes through it, this shorts out the top screen and will shut the DS down

you need a new case and top LCD as the cable running through the hinge is fragile and has obviously been broken. Ebay should provide you with spares required and youtube for video's on how to do the job

The right hand side hinge has the ribbon cable for the top screen inside and the 2 wires for ths mic and wireless, chances are the ribbon cable has become damaged, therefore the top screen will proberly need replacing.

sorry to say that your top screen needs completely replacing if it is doing as described!!
the bottom screen and the touch screen are seperate, jus mounted on top of each other, so if the picture works on the bottom then the bottom screen is fine just the touch screen is broken. again i think if the cables/ribbons are in correctly and it isn't responding then it will need replacing!!

You can buy a new plastic housing for your DS lite on many different sites. Some are great quality. If the DS still works (despite the broken hinge) it makes sense to replace the housing.
You should be able to find enough videos on youtube to learn how to disassembe the DS and not do any harm. Make sure that you buy a tri-wing screwdriver (also called a nintendo screwdriver).

There is a ribbon that connects the top screen of the ds to the motherboard. It is possible that this ribbon became damaged when the hinge broke. A possible fix would be to replace the upper screen. Hope this helps.